Lecture 13 - Multifactor ANOVA
Lecture 12: Review
ANOVA
- Analysis of variance: single and multi-factor designs
- Examples: diatoms, circadian rhythms
- Predictor variables: fixed vs. random
- ANOVA model
- Analysis and partitioning of variance
- Null hypothesis
- Assumptions and diagnostics
- Post F Tests - Tukey and others
- Reporting the results
Lecture 13: Multifactor ANOVA Overview
Multifactor ANOVA
- nested and factorial designs
Nested design examples
- Factorial design examples
- Nested designs
- Linear model
- Analysis of variance
- Null hypotheses
- Unbalanced designs
- Assumptions
Lecture 13: 2 Factor or 2 Way ANOVA
Often consider more than 1 factor (independent categorical variable):
- reduce unexplained variance
- look at interactions
2-factor designs (2-way ANOVA) very common in ecology
- Can have more factors (e.g., 3-way ANOVA)
- interpretation tricky…
Most multifactor designs: nested or factorial
Lecture 13: Nested and factorial designs
Consider two factors: A and B
- Nested/hierarchical: levels of B occur only in 1 level of A
- Factorial/crossed: every level of B in every level of A
Lecture 13: Nested and factorial designs
Nested Designs:
- Factor A usually fixed
- Factor B usually random
Lecture 13: Nested and factorial designs
Factorial Designs:
- Both factors typically fixed (but not always)
Lecture 13: Nested designs: examples
Study on effects of enclosure size on limpet growth:
- 2 enclosure sizes (factor A)
- 5 replicate enclosures (factor B)
- 5 replicate limpets per enclosure
Lecture 13: Nested designs: examples
Study on reef fish recruitment: 5 sites (factor A) 6 transects at each site (factor B) replicate observations along each transect
Lecture 13: Nested designs: examples
Effects of sea urchin grazing on biomass of filamentous algae:
- 4 levels of urchin grazing: none, L, M, H
- 4 patches of rocky bottom (3-4 m2) nested in each level of grazing
- 5 replicate quadrats per patch
F
Lecture 13: Factorial designs: examples
Effects of light level on growth of seedlings of different size:
- 3 light levels (factor A)
- 3 size classes (factor B)
- 5 replicate seeding in each cell
Lecture 13: Factorial designs: examples
Effects of food level and tadpole presence on larval salamander growth
- 2 food levels (factor A)
- presence/absence of tadpoles (factor B)
- 8 replicates in each cell
Lecture 13: Factorial designs: examples
Effect of season and density on limpet fecundity.
- 2 seasons (factor A)
- 4 density treatments (factor B)
- 3 replicates in each cell
F
Lecture 13: Nested designs: linear model
Consider a nested design with:
- p levels of factor A (i= 1…p) (e.g., 4 grazing levels)
- q levels of factor B (j= 1…q), nested within each level of A (e.g., 4 - diff. patches per grazing level)
- n replicates (k= 1…n) in each combination of A and B (5 replicate - quadrats in each patch in each grazing level)
I
Lecture 13: Nested designs: linear model
Can calculate several means:
- overall mean (across all levels of A and B)= ȳ;
- a mean for each level of A (across all levels of B in that A)= ȳi;
- a mean for each level of B within each A= ȳj(i)
Lecture 13: Nested designs: linear model
Lecture 13: Nested designs: linear model
The linear model for a nested design is: \[y_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{ijk}\]
Where:
\(y_{ijk}\) is the response variable
value of the k-th replicate in j-th level of B in the i-th level of A
(algal biomass in 3rd quadrat, in 2nd patch in low grazing treatment)
\(\mu\) is the overall mean
- (overall average algal biomass)
Lecture 13: Nested designs: linear model
The linear model for a nested design is:
The linear model for a nested design is: \[y_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{ijk}\]
\(\alpha_i\) is the fixed effect of factor \(i\)
(difference between average biomass in all low grazing level quadrats and overall mean)
\(\beta_{j(i)}\) is the random effect of factor \(j\) nested within factor \(i\)
usually random variable, measuring variance among all possible levels of B within each level of A
(variance among all possible patches that may have been used in the low grazing treatment)
Lecture 13: Nested designs: linear model
The linear model for a nested design is:
The linear model for a nested design is: \[y_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{ijk}\]
- \(\varepsilon_{ijk}\) is the error term
- αi: is the effect of the ith level of A: µi- µ
- unexplained variance associated with the kth replicate in jth level of B in the ith level of A
- (difference bw observed algal biomass in 3rd quadrat in 2nd patch in low grazing treatment and predicted biomass - average biomass in 2nd patch in low grazing treatment)
Lecture 13: Nested designs: analysis of variance
As before, partition the variance in the response variable using SS SSA is SS of differences between means in each level of A and overall mean
Lecture 13: Multifactor ANOVA
SSB is SS of difference between means in each level of B and the mean of corresponding level of A summed across levels of A
Lecture 13: Nested designs: analysis of variance
- SSresid is difference bw each observation and mean for its level of factor B, summed over all observations
- SStotal = SSA + SSB + SSresid
- SS can be turned into MS by dividing by appropriate df
Lecture 13: Nested designs: analysis of variance
Lecture 13: Nested designs: null hypotheses
Two hypotheses tested on values of MS:
- no effects of factor A
- Assuming A is fixed:
- Ho(A): µ1= µ2= µ3=…. µi= µ
- Same as in 1-factor ANOVA, using means from B factors nested within each - level of A
- (no difference in algal biomass across all levels of grazing: µnone= - µlow= µmed= µhigh)
Lecture 13: Nested designs: null hypotheses
Two hypotheses tested on values of MS:
- No effects of factor B nested in A
- Assuming B is random:
- Ho(B): σβ2= 0 (no variance added due to differences between all possible - levels of B)
- (no variance added due to differences between patches)
Lecture 13: Nested designs: null hypotheses
Conclusions?
“significant variation between replicate patches within each treatment, but no significant difference in amount of filamentous algae between treatments”
Lecture 13: Nested designs: unbalanced designs
Unequal sample sizes can be because of:
- uneven number of B levels within each A
- uneven number of replicates within each level of B
Not a problem, unless have unequal variance or large deviation from - normality
Lecture 13: Nested designs: assumptions
As usual, we assume
- equal variance
- normality
- independence of observations
Equal variance + normality need to be assessed at both levels:
- Since means for each level of B within each A are used for the H-test about A, need to assess whether those means meet normality and equal variance
- Examine residuals for H-test about B
- Transformations can be used